Tree embeddings and nonuniqueness in site percolation

TL;DR

This paper proves a nonuniqueness theorem for Bernoulli site percolation on properly embedded planar graphs with minimum degree at least 7, establishing multiple infinite clusters in a certain probability range and introducing new embedded-tree separation techniques.

Contribution

It introduces an explicit embedded-tree separation mechanism for planar nonuniqueness and extends results to general graphs beyond planarity.

Findings

Proves nonuniqueness of infinite clusters for certain percolation probabilities.

Constructs embedded trees and forests with separation properties leading to exponential decay.

Establishes a lower bound on two-point connectivity under uniqueness for arbitrary graphs.

Abstract

We prove a nonuniqueness theorem for Bernoulli site percolation on properly embedded planar graphs, and we obtain a general connectivity principle beyond planarity. Let $G$ be an infinite connected graph properly embedded in $\RR^2$ with minimum degree at least $7$. Then \[ p_c^{\mathrm{site}}(G)<\tfrac12, \] and for every \[ p\in \bigl(p_c^{\mathrm{site}}(G),\,1-p_c^{\mathrm{site}}(G)\bigr), \] Bernoulli$(p)$ site percolation on $G$ has almost surely infinitely many infinite open clusters. In particular, this verifies a conjecture of Benjamini and Schramm for properly embedded planar graphs. The core new ingredient is an explicit embedded-tree separation mechanism for planar nonuniqueness. We construct embedded trees and an embedded forest whose separation properties yield exponential decay of two-point connection probabilities in the matching graph. To treat the high-density regime, we introduce a binary-tree version of uniform percolation and prove stability of infinite clusters under edge additions, without any bounded-degree assumption. Beyond the planar theorem, we prove a general lower bound on two-point connectivity under uniqueness for arbitrary infinite locally finite graphs. As a consequence, if \[ p_c^{\mathrm{site}}(G)<p<p_{\mathrm{conn}}(G), \] then Bernoulli site percolation on $G$ has almost surely infinitely many infinite open clusters.